Week 4: Exponential and Logarithmic Functions

Course: Jan 2026 - Mathematics I Difficulty: Moderate Focus: Growth, decay, inverse functions, and their powerful properties.

1. Exponential Functions

• $a$ is a positive real number ($a > 0$) and $a \neq 1$. $a$ is called the base.
• $x$ is any real number. $x$ is called the exponent.

1.1 Behaviours and Graphs

• Exponential Growth ($a > 1$): As $x$ increases, $y$ increases rapidly. The curve goes upwards from left to right. Example: Population growth.
• Exponential Decay ($0 < a < 1$): As $x$ increases, $y$ decreases, approaching zero but never touching it. The curve goes downwards from left to right. Example: Radioactive decay.

1.2 The Natural Exponential Function

1.3 Laws of Exponents

1. $a^x \cdot a^y = a^{x+y}$
2. $\frac{a^x}{a^y} = a^{x-y}$
3. $(a^x)^y = a^{xy}$
4. $(ab)^x = a^x b^x$
5. $a^{-x} = \frac{1}{a^x}$

2. Logarithmic Functions

• Base $a > 0$ and $a \neq 1$.
• Argument $x > 0$. (We cannot take the logarithm of zero or a negative number).

2.1 Graph of a Logarithmic Function

• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$
• The $y$-axis is a vertical asymptote.
• Passes through $(1, 0)$ because $\log_a(1) = 0$.

2.2 Common Logarithms

• Common Logarithm: Base 10. Usually written as $\log(x)$ or $\log_{10}(x)$.
• Natural Logarithm: Base $e$. Written as $\ln(x)$. Thus, $\ln(x) = y \iff e^y = x$.

3. Properties of Logarithms

1. Product Rule: $\log_a(MN) = \log_a(M) + \log_a(N)$
2. Quotient Rule: $\log_a(\frac{M}{N}) = \log_a(M) - \log_a(N)$
3. Power Rule: $\log_a(M^p) = p \cdot \log_a(M)$
4. Change of Base Formula: $\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$ (often used to convert to base 10 or base $e$ for calculation).

4. Exponential and Logarithmic Equations

4.1 Solving Exponential Equations

• Same Base Method: If you can express both sides with the same base, equate the exponents. If $a^u = a^v$, then $u = v$.
• Taking Logarithms: If bases differ, take the logarithm (usually $\ln$ or $\log$) of both sides and use the power rule to bring the variable exponent down. $2^x = 7 \implies \ln(2^x) = \ln(7) \implies x \ln(2) = \ln(7) \implies x = \frac{\ln(7)}{\ln(2)}$

4.2 Solving Logarithmic Equations

1. Condense: Use log properties to combine multiple logs into a single logarithm on each side.
2. Exponentiate: Convert the equation from logarithmic form to exponential form. $\log_a(x) = y \implies x = a^y$.
3. Check for Extraneous Solutions: ALWAYS plug your potential solutions back into the original equation to ensure the arguments of all logarithms are strictly positive. Solutions that produce $\log(\text{negative})$ or $\log(0)$ must be rejected.

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